When viewed graphically, the games \(\texttt {REL}\)  and \(\texttt {RAV}\)  could be re-phrased as the make a cycle game and the avoid a cycle game, respectively. Hence, one could think of these combinatorial games in a purely graph-theoretic sense. However, \(\texttt {REL}\)  and \(\texttt {RAV}\)  are distinctly different from the combinatorial games involving graphs found in the current literature.
For example, the game Cops and Robbers (see [1]}, [2]}) is a pursuit-evasion game and has been studied specifically on Cayley graphs as well (see [3]}). Although the games \(\texttt {REL}\)  and \(\texttt {RAV}\)  involve evasion of some kind, one would not classify them as a pursuit-evasion game like Cops and Robbers. Neither player is trying to capture the other and our games disallow backtracking.
